होम Journal of Philosophical Logic The Logic of Pragmatic Truth

The Logic of Pragmatic Truth

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December, 1998
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In memory of Rolando Chuaqui

ABSTRACT. The mathematical concept of pragmatic truth, first introduced in Mikenberg,
da Costa and Chuaqui (1986), has received in the last few years several applications in
logic and the philosophy of science. In this paper, we study the logic of pragmatic truth,
and show that there are important connections between this logic, modal logic and, in
particular, Jaskowski’s discussive logic. In order to do so, two systems are put forward so
that the notions of pragmatic validity and pragmatic truth can be accommodated. One of
the main results of this paper is that the logic of pragmatic truth is paraconsistent. The
philosophical import of this result, which justifies the application of pragmatic truth to
inconsistent settings, is also discussed.
KEY WORDS: Jaskowski’s logic, modal logic, paraconsistent logic, partial structures,
pragmatic truth, quasi-truth

In Mikenberg, da Costa and Chuaqui (1986), the mathematical concept of
pragmatic truth was introduced, an infinitary logical system was presented
to treat this concept, and some applications of it made in logic and in
algebra. An application of this concept in the foundations of the theory of
probability is studied in da Costa (1986), and some extensions of it to inductive logic and to the philosophy of science are considered, respectively,
in da Costa and French (1989), and da Costa and French (1990). This
framework was also employed in order to examine some issues involved in
the theory of acceptance (da Costa and French (1993a)), as well as in the
modelling of ‘natural reasoning’ (da Costa and French (1993b)).
The wide range of applications of the notion of pragmatic truth motivates an investigation of the logic of pragmatic truth. In this paper, we will
show that there are important connections between this logic and modal
logic, in particular between the logic of pragmatic truth and Jaskowski’s
discussive logic (see Jaskows; ki (1969) and da Costa (1975)).2 We will
study these connections in the context of da Costa’s formulation of pragmatic structures (cf. da Costa (1986)), which we shall present in Section 1.
As we shall see in Section 2, such structures can be considered as worlds
Journal of Philosophical Logic 27: 603–620, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.

VTEX(VR) PIPS No.: 154813 (logikap:humnfam) v.1.15
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of a Kripke structure, and in this setting the necessity operator in modal
logic corresponds to the notion of pragmatic validity, and the possibility
operator to the notion of pragmatic truth. Two systems are then put forward
in order to study these notions (respectively, in Sections 2 and 3). One of
the main results of the present paper, proved in Section 3, is that the logic of
pragmatic truth is paraconsistent. The philosophical import of this result,
which justifies the application of pragmatic truth to inconsistent settings,
is then discussed in Section 4.
A remark on our terminology is important here. We call the kind of truth
defined in this paper pragmatic truth, owing to its connections with the
pragmatic conception of truth, as developed by philosophers like James,
Dewey and particularly Peirce (cf. Mikenberg etal. (1986), da Costa (1986),
and da Costa and French (forthcoming), Chapter 1). However, our piece is
not exegetical. The sole point we would like to emphasize is that our definition was heuristically inspired by some passages of pragmatic thinkers,
such as Peirce, when he wrote that, “consider what effects, that might
conceivably have practical bearings, we conceive the object of our conceptions to have. Then, our conception of these effects is the whole of our
conception of the object” (Peirce (1965), p. 31).
In our opinion, the definition of pragmatic truth studied in this paper captures, at least in part, the common concept of a theory saving the
appearances, usually by means of partially fictitious constructions (see
Vaihinger (1952), and Bueno (1997)). Maybe it would be better to call our
kind of truth quasi-truth, instead of pragmatic truth.

In order to motivate the study of the set-theoretical structures which we
shall call here simple pragmatic structures, we begin with some informal
Let us suppose that we are interested in studying a certain domain of
knowledge 1 in the field of empirical sciences, for instance the physics of
particles. We are, then, concerned with certain real objects (in the physics
of particles, with some configurations in a Wilson chamber, some spectral
lines etc.), whose set we denote by A1 . Among the objects of A1 , there
are some relations that interest us and that we model as partial relations
Ri , i ∈ I , every relation having a fixed arity. The relations Ri , i ∈ I ,
are partial relations, that is, each Ri , supposed of arity ri , is not necessarily defined for all ri -tuples of elements of A1 , i ∈ I .3 The reason
for using partial relations is that they are supposed to express what we
do know, or what we accept as true, about the actual relations linking

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the elements of A1 . Then, the partial structure hA1 , Ri ii∈I encompasses,
so to say, what we know or accept as true about the actual structure of
1 (for further formal details, see Mikenberg et al. (1986) and da Costa
However, in order to systematize our knowledge of 1, it is convenient
to introduce in our structure hA1 , Ri ii∈I some ideal objects (in the physics
of particles, quarks for example). The set of these new objects will be
denoted by A2 . It is understood that A1 ∩A2 = ∅, and we put A = A1 ∪A2 .
This way, the modelling of 1 involves new partial relations Rj , j ∈ J ,
some of which extend the relations Ri , i ∈ I .
Furthermore, there are some sentences (closed formulas) of the language L, in which we talk about the structure hA, Rk ik∈I ∪J (I ∩ J = ∅)
that we accept as true or that are true (in the sense of the correspondence
theory of truth). This occurs, for instance, with sentences expressing true
decidable propositions (a proposition whose truth or falsehood can be decided), and with some general sentences which express laws or theories
already accepted as true. Let us denote the set of such sentences, dubbed
primary, by P .
Then, taking into account the above informal discussion, we are led
to suggest that a simple pragmatic structure be regarded as a set-theoretic
structure of the form
A = hA1 , A2 , Ri , Rj , P ii∈I,j ∈J
where the nonempty sets A1 and A2 , the partial relations Ri and Rj , i ∈ I
and j ∈ J , and the set of sentences P satisfy the preceding conditions.
More formally, such a structure can be defined as follows:
DEFINITION 1. A simple pragmatic structure (sps) is a structure
A = hA, Rk , P ik∈K ,
where A is a nonempty set, Rk is a partial relation defined on A for every
k ∈ K, and P is a set of sentences of a language L of the same similarity
type as that of A and which is interpreted in A. The set A is called the
universe of A. (For some k, Rk may be empty; P may also be empty.)
From now on, a sps will always be a structure according to Definition 1.
(In order to simplify the exposition, partially defined functions are not
included in a sps.)
Let L be a first-order language with equality, but without function symbols. The symbols of L are, then, logical symbols (connectives, individual variables, the quantifiers, and the equality symbol), auxiliary symbols

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(parentheses), a collection of individual constants, and a collection of predicate symbols. To interpret L in a sps A = hA, Rk , P ik∈K is to associate
with each individual constant of L an element of the universe of A, and
with each predicate symbol of L of arity n a relation Rk , k ∈ K, of the
same arity. It is supposed that every predicate of the family (Rk )k∈K is
associated with a predicate symbol.
DEFINITION 2. Let L and A = hA, Rk , P ik∈K be respectively a language and a sps, such that L is interpreted in A. Let S be a total structure,
that is a usual structure (whose relations of arity n are defined for all
n-tuples of elements of its universe), and we suppose that L is also interpreted in S. Then, S is said to be A-normal if the following properties
are verified:
(1) The universe of S is A.
(2) The (total) relations of S extend the corresponding partial relations
of A.
(3) If c is an individual constant of L, then in both A and S c is interpreted
by the same element.
(4) If α ∈ P , then S |= α.
Given a pragmatic structure A, it may happen that there are no A-normal
structures. It is possible to provide a system of necessary and sufficient
conditions for the existence of A-normal structures (see Mikenberg et al.
(1986), and da Costa (1974)). One condition of this system is as follows.
For each partial relation Rk in A, we construct a set Mk of atomic sentences
and negations of atomic sentences such that the former corresponds to ntuples that satisfy Rk , and the latter to n-tuples that do not satisfy Rk (such
S correspond to n-tuples in the “anti-extension” of Rk ). Let M be
the set k∈K Mk . Therefore, a sps A admits an A-normal structure only if
the set M ∪ P is consistent. In what follows we shall always suppose that
our sps satisfy the relevant conditions; in other words, given any sps A, the
set of A-normal structures is not empty.
DEFINITION 3. Let A be respectively a language and a sps in which L
is interpreted. We say that a sentence α is pragmatically true in the sps A
according to S if
(1) A is a sps,
(2) S is an A-normal structure, and
(3) α is true in S (in conformity with the Tarskian definition of truth).
That is, we say that α is pragmatically true in the sps A if there exists
an A-normal S in which α is true (in the Tarskian sense). If α is not

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pragmatically true in the sps A according to S (is not pragmatically true in
the sps A), we say that α is pragmatically false in the sps A according to
S (is pragmatically false in the sps A).


Given a sps A, it is natural to consider its A-normal structures as the
worlds of a Kripke structure for S5 with quantification, i.e., we have a
universe and several structures, defined in such a universe, in which the
language L can be interpreted, and where every world is accessible to every
world (cf. Hughes and Cresswell (1968)). It is also natural to extend the
language L of the sps A to a modal language, by the adjunction of the
modal operator  to its primitive symbols. The operator  which in modal
logic represents the notion of necessity, corresponds in the present situation
to pragmatic validity (in a sps A). Analogously, the possibility symbol ♦,
definable in terms of  and negation, corresponds to pragmatic truth (in a
sps A). Thus, we are led to extend the semantics of L in an obvious way,
such that the symbols  and ♦ will represent the concepts of pragmatic
validity and of pragmatic truth, respectively. Moreover, since the universes
of all “worlds” belonging to a sps are the same, it is reasonable that equality behaves, in the cases of pragmatic truth and of pragmatic validity, as
necessary equality (cf. Hughes and Cresswell (1968), Chapter 11).
Among the pragmatically valid formulas – that is, those formulas α
such that α is a theorem of S5 with quantification and necessary equality −, there are the logically pragmatically true formulas – that is, those
formulas α such that ♦α or, equivalently, ♦α is a theorem of the same
system. From now on, in order to simplify the language, the former class
of formulas will be called strictly pragmatically valid and the latter will
be called pragmatically valid. The first class of formulas coincides with
the set of theorems of S5 with quantification and necessary equality; the
second, with Jaskowski’s logic associated with the same system. (We recall that, loosely speaking, given a modal system M, the Jaskowski’s logic
associated to M is the set of all formulas a such that ♦α is a thesis of M;
see Jaskowski (1969), Kotas and da Costa (1977) and da Costa (1975).)4
Therefore, the logical system which will be denoted by P V and which
systematizes the notion of strict pragmatic validity has a language L∗ whose
primitive symbols are those of a standard formalization of the first-order
predicate calculus with equality and individual constants, plus the symbol
 (for simplicity, function symbols are excluded). The defined symbols
are introduced as usual, and the common conventions in the writing of for-

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mulas, and in the formulation of postulates (axiom schemes and primitive
rules of inference) etc. are employed without explicit mention.
The postulates of P V are the following:

If α is an instance of a (propositional) tautology, the α is an axiom.
α, α → β/β
(α → β) → (α → β)
α → α
♦α → ♦α
∀xα(x) → α(t)
α → β(x)/α → ∀xβ(x)
x = y → (α(x) ↔ α(y))

In the postulates above, the symbols have clear meanings. In particular, in
Axiom Scheme 6, t is either a variable free for x in α(x) or an individual
constant. As we have already noticed, this system is essentially S5.
We define the concept of deduction as in Henkin and Montague (1956).
Their basic idea is essentially that one can only use the generalization
rule – α/∀xα – in a step k of a deduction when a subsequence of the
deduction up to k is a proof of α. This restriction to the generalization
rule is exactly similar to the one adopted with regard to the necessitation
rule, α/α. Then, the usual derived rules, such as the deduction theorem,
remain valid.5
The semantics of P V can be easily developed: the basic (strict) semantical concepts of pragmatic truth, pragmatic falsehood, pragmatic validity,
pragmatic invalidity, pragmatic semantic consequence etc. offer no difficulties in being formulated (and maintain the spirit of Definition 3). This
claim notwithstanding, we reproduce here the definition of satisfiability.
Let L∗ be a language and A a sps in which L∗ is interpreted. (L∗ is a
modal first-order language.) So, to each individual constant c of L∗ there
is associated an element cA of the universe of A, and to each predicate
symbol P of arity n of L∗ there is associated an n-ary partial relation PA ,
defined on the same universe. Moreover, with an A-normal structure S
fixed, to each n-ary predicate symbol P of L∗ there is associated an n-ary
total relation PS , and to each individual constant c of L∗ corresponds a
member cS of the universe of S, such that cA = cS .
An assignment (of L∗ in A) is a function f whose domain is the collection of individual variables of L∗ , which maps to each variable x an
element f (x) of the universe of A.

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If f is an assignment of L∗ in A, then f 0 is the function defined on
the set of all variables and constants of L∗ , such that f 0 (t) is tA if t is a
constant, and f 0 (t) is f (t) if t is a variable.
Let A be a sps whose language is L∗ , and f be an assignment of L∗ in
A; also let S be an A-normal structure. We define the relation
hA, f, Si |= α,
or, in words, hA, f, Si satisfies the formula α of L∗ , by recursion on α:
(1) If P is a predicate symbol of arity n and t1 , t2 , . . . , tn are n terms, then
hA, f, Si |= P (t1 , t2 , . . . , tn ) iff hf 0 (t1 ), f 0 (t2 ), . . . , f 0 (tn )i ∈ PS .
(2) If ti and tj are two arbitrary terms, then hA, f, Si |= ti = tj iff
f 0 (ti ) = f 0 (tj ).
(3) The common satisfaction conditions for the primitive connectives and
the primitive quantifier.
(4) If β is a formula, then hA, f, Si |= β iff hA, f, Si |= β, for all
A-normal structure S.
We have the following theorem, whose proof can be obtained by the methods of Hughes and Cresswell (1968) or of Gallin (1975):
THEOREM 1. Let 0 be a set of formulas of L∗ and α be a formula of the
same language. Then, 0 ` α if, and only if, 0 |= α (where ` and |= have
clear meanings).
That is, α is a syntactic consequence of 0 in P V iff α is a strict pragmatic semantic consequence of 0.
The logic of strict pragmatic validity, which we have just sketched,
can be extended to higher-order (modal) languages, for example by an
adaptation of some ideas presented in Gallin (1975), Chapter 3. Moreover,
one can also develop a metatheoretical study of this logic, for example,
by adopting different modal systems as basic (S4, for instance), by distinguishing between frames and models etc. Instead of pursuing this line
here, we will now consider a different system in order to study the logic
of pragmatic validity. This system, as we will see, is constructed, as it
were, in terms of P V , which was presented here mainly as an auxiliary
construction. We will then show that the logic of pragmatic validity is

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The system P V constitutes a formalization of the notion of strict pragmatic
validity. Now we present a system P T , whose language is the same as
that of P V , which constitutes a formalization of the notion of pragmatic
validity; the underlying intuition is that of constructing a system in which
` α means that ♦α is strictly pragmatically valid.
Suppose that α is a formula of L∗ , the language of P V or of P T . We
denote by ∀∀α any formula of the form
∀x1 ∀x2 . . . ∀xn α,
where any free variable of α is one of the variables x1 , x2 , . . . , xn . (There
may be variables in the sequence x1 , x2 , . . . , xn , which are not free in α.)
When n = 0, we take ∀∀α to be α.
Clearly, in order that α be pragmatically valid in the sense intended, we
must have that ` α in P T if, and only if, ` ♦∀∀α in P V . So, P T is a
kind of Jaskowski’s discussive logic associated with P V (see Jaskowski
(1969), da Costa (1975), and Kotas and da Costa (1977)). By definition,
then, α is a theorem of P T iff ♦α is a theorem of P V , and this definition
directly furnishes a semantical interpretation for P T .
P T can be axiomatized as follows:
(10 ) If α is an instance of a (propositional tautology), then ∀∀α is an
(2 ) ∀∀α, ∀∀(α → β)/∀∀β
(30 ) ∀∀((α → β) → (α → β))
(40 ) ∀∀(α → α)
(50 ) ∀∀(♦α → ♦α)
(60 ) ∀∀(∀xα(x) → α(t))
(70 ) ∀∀α/α
(80 ) ∀∀α/∀∀α
(90 ) ♦∀∀α/α
(100 ) ∀∀(α → β(x))/∀∀(α → ∀xβ(x))
(110 ) Vacuous quantifications may be introduced or suppressed in any
(12 ) ∀∀x(x = x)
(130 ) ∀∀(x = y → (α(x) ↔ α(y)))
In P T the definitions of proof and of (formal) theorem are the usual

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We proceed to show that postulates (10 )–(130 ) provide an axiomatization
for P T .
LEMMA 1. If α is a theorem of our proposed axiomatization for P T , then
♦∀∀α is a theorem of P V .
Proof. By induction on the length of the proof of α in the proposed axiomatization for P T .
Let α1 , α2 , . . . , αn , where αn is α, be a (formal) proof of α in the proposed axiomatization for P T . Then, αi , 1 6 i 6 n, is an axiom or is
obtained by the application of one of the rules.
If αi is an axiom, then it has the form ∀∀β, where β is an axiom of
P V (observe that P V is S5 with quantification and necessary equality).
Therefore, ∀∀β is a theorem of P V , and so ♦∀∀∀∀β, i.e., ♦∀∀αi is
also a theorem of P V .
Suppose that αi is a consequence of two preceding formulas by Rule 20 .
Then αi is ∀∀β, obtained from the premises ∀∀γ and ∀∀(γ → β).
By the induction hypothesis, ♦∀∀∀∀γ and ♦∀∀∀∀(γ → β) are provable in P V . Consequently, ∀∀γ and ∀∀(γ → β) are also provable in
P V , and so is ∀∀β. But if ∀∀β is a theorem of P V , then, ♦∀∀∀∀β,
i.e., ♦∀∀αi , is also a theorem. The other rules are similarly treated.
LEMMA 2. If α is a theorem of P V , then ∀∀α is a theorem of the
proposed axiomatization for P T .
By induction on the length of the proof of α in P V .
Let α1 , α2 , . . . , αn , where αn is α, be a proof of α in P V .
If αi , 1 6 i 6 n, is an axiom of P V , then ∀∀α is a theorem of the
proposed axiomatization for P T , as is obvious. If αi is obtained by an
application of modus ponens (Rule 2), from γ and γ → αi , we have, by
the induction hypothesis, that ∀∀γ and ∀∀(γ → αi ) are provable in the
proposed axiomatization. Then, by Rule 2, ∀∀αi is also provable. Rule 8
is treated analogously.
THEOREM 2. Postulates (10 )–(130 ) characterize P T ; that is, we have:
` α in P T iff ` ♦∀∀α in P V .
Proof. Let us suppose that α is a theorem of the proposed axiomatization
for P T ; then, by Lemma 1, ♦∀∀α is a theorem of P V .

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Assume that ♦∀∀α is a theorem of P V . So, by Lemma 2, ∀∀♦∀∀α is
a theorem of the proposed axiomatization for P T . Therefore, by Rule 70 ,
♦∀∀α is a theorem of P T , and so by Rule 90 , α is also a theorem of P T . 2
DEFINITION 4. In P T we say that the formula α is a syntactic consequence of a set of formulas 0 (in symbols, 0 ` α) if there exist γ1 , γ2 , . . . ,
γn in 0, such that
(♦γ1 ∧ ♦γ2 ∧ · · · ∧ ♦γn ) → ♦α
is a theorem of P T (or, equivalently,
♦∀∀((♦γ1 ∧ ♦γ2 ∧ · · · ∧ ♦γn ) → ♦α)
is a theorem of P V ). When n = 0, the first formula above reduces, by
convention, to α (and ∅ ` α means, thus, that ` α).
DEFINITION 5. A pragmatic theory is a set T of sentences (closed formulas of P T ), such that if γ1 , γ2 , . . . , γn are in T and {γ1 , γ2 , . . . , γn } ` α,
then α is also in T .
THEOREM 3. If T is a pragmatic theory and α is a (closed) theorem of
P T , then α ∈ T .
DEFINITION 6. Let E be the set of all sentences of P T and T be a
pragmatic theory. T is called trivial (or overcomplete) if T = E; otherwise,
T is called nontrivial. The theory T is called inconsistent if there is at least
one sentence α such that α ∈ T and ¬α ∈ T , where ¬ is the symbol of
negation of P T ; otherwise, T is called consistent.6
THEOREM 4. There exist pragmatic theories which are inconsistent but
Proof. Let c and M be respectively any individual constant and a monadic
predicate symbol of P T . The theory whose (nonlogical) axioms are M(c)
and ¬M(c) is inconsistent. But it is nontrivial, because the corresponding
theory of P V , whose (nonlogical) axioms are ♦M(c) and ♦¬M(c), is
consistent. In effect, it is easy to construct a Kripke model for P V in which
both ♦M(c) and ♦¬M(c) are true. However, in no Kripke model for P V
the formula ♦(M(c) ∧ ¬M(c)) is true. (See also Theorem 9 of da Costa
DEFINITION 7. If α and β are formulas, we put
α →d β stands for ♦α → β;
α ∧d β stands for ♦α ∧ β.

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The connectives →d and ∧d are called discussive implication and discussive conjunction, respectively.
THEOREM 5. In P T →d , ∧d and ∨ satisfy all the valid schemes and
rules of classical positive logic.
Proof. If we consider a valid primitive scheme (or rule) of classical positive
logic, and replace in it implication by discussive implication and conjunction by discussive conjunction, we obtain a valid scheme (or rule) of P T ,
as is easily seen.
THEOREM 6. If T is a pragmatic theory, then α ∈ T iff there exist
γ1 , γ2 , . . . , γn in T such that
(γ1 ∧d γ2 ∧d · · · ∧d γn ) →d ♦α
is a theorem of P T .
Da Costa (1975) contains a different formulation of the discussive logic
associated with P V , which we will dub here P T ∗ . P T ∗ is defined as
follows: α is a theorem of P T ∗ iff ♦α is a theorem of P V . It is clear
that our definition of the discussive logic associated with P V is better than
P T ∗ from the point of view of pragmatic truth. We have:
THEOREM 7. If α is a theorem of P T , then α is also a theorem of P T ∗ .
Proof. In effect, if ` α in P T , then, by definition, ` ♦∀∀α in P V ; but in
P V we have:
` ♦∀∀α → ∀∀♦α.
Hence, in P V , ` ∀∀♦α, and ` ♦α, since in the last system the scheme
∀∀α → α is valid.
THEOREM 8. There are theorems of P T ∗ which are not theorems of P T .
Proof. The proof consists in the presentation of a formula α such that ` α
in P T ∗ , i.e., ` ♦α in P V , but it is not the case that ` α in P T , i.e. it is
not the case that ` ♦∀∀α in P V .
Let α be the formula ♦M(x) → M(x), where M is a monadic predicate
symbol. We have:
(1) Since ♦(♦p → p), where p is a propositional variable, is a theorem
of S5, we deduce that in P V
` ♦(♦M(x) → M(x)).

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Therefore, ♦M(x) → M(x) is a theorem of P T ∗ .
(2) It is not the case that ` ♦∀∀x(♦M(x) → M(x)) in P V . The verification of this assertion consists of obtaining a Kripke structure for P V
in which the following formula becomes true:
¬♦∀x(♦M(x) → M(x))
that is
∃x(♦M(x) ∧ ¬M(x)).
We consider a Kripke model hW, D, V i (see Hughes and Cresswell
(1968)), where W = {w1 , w2 }, D = {a, b}, V (M, w1 ) = {a} and V (M,
w2 ) = {b}. This structure satisfies the preceding formula.
We observe that in the applications of the theory developed above, it is
sometimes convenient to employ an alternative definition of syntactic consequence. For instance, in certain applications of our ideas in the domain
of the foundations of physics, instead of Definition 4, it is more appropriate
to adopt the following one:
DEFINITION 40 . In P T , the sentence α is said to be a proper syntactic
consequence of a set of sentences 0 if there is γ1 , γ2 , . . . , γn in 0, such
♦(γ1 ∧ γ2 ∧ · · · ∧ γn ) and ((γ1 ∧ γ2 ∧ · · · ∧ γn ) → γ )
are theorems of P T (or of P T and some extra axioms).
The corresponding logic developments and applications will be treated
in forthcoming works.

The philosophical significance of the above formal account can be drawn
out through a consideration of inconsistency in our belief systems. If one
focuses on Theorem 4, it can be seen that a pragmatic theory can contain contradictory theorems without reducing to triviality. This means that
P T belongs to the class of paraconsistent logics, since a logical system is
paraconsistent if it can be employed as the underlying logic of inconsistent
but nontrivial deductive systems or theories (for a useful outline of the nature of paraconsistent logics, see Arruda (1980)). In the case of pragmatic
truth, this is not an unreasonable situation: contradictory propositions may,
of course, both be pragmatically true. Thus pragmatic truth can be used
to provide the epistemic framework for characterizing inconsistent belief

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The occurrence of inconsistent theories in science, for example, is now
a widely recognized phenomenon; examples range from Bohr’s theory of
the atom and the “old” quantum theory of black-body radiation to the
infinitesimal calculus and Stokes’ analysis of the motion of a pendulum.
The problem, of course, is how to accommodate this aspect of scientific
practice given that within the framework of classical logic an inconsistent
result is disastrous: the set of consequences of an inconsistent theory will
explode into triviality and the theory is rendered useless. Another way of
expressing this descent into logical anarchy is to say that under classical
logic the closure of any inconsistent set of sentences includes every sentence. It is this which lies behind Popper’s famous declaration that the
acceptance of inconsistency “[. . .] would mean the complete breakdown
of science” (Popper (1940)).
Various approaches for accommodating inconsistent theories can be
broadly delineated. Thus one can distinguish “logic driven control” of this
apparent logical anarchy from “content driven control”, where the former
involves the adoption of some underlying non-classical logic and the latter
focuses on the specific content of the theory in question (Norton (1993)).
A well known example of the former approach is the work of Priest,
who cites the existence of inconsistency in science as the pragmatic “fulcrum” of his approach, levering the reader into acceptance of the Hegelian
view that there exist true contradictions, where “true” is here understood
in the correspondence sense (Priest (1987)). The classical collapse of an
inconsistent theory into triviality is then prevented through the adoption
of a form of paraconsistent logic in which certain contradictions are tolerated. That such an approach fails to do justice to the doxastic attitude
of the scientists developing such theories is clear: no-one believed that
Bohr’s theory was true, not even Bohr himself (see da Costa and French
According to he alternative “content driven” approach, the attitude of
scientists to inconsistency is based simply on “[. . .] a reflection of the specific content of the physical theory at hand” (Norton (1993), p. 417). Thus,
it is argued, from an inconsistent theory we can construct a consistent subtheory yielding the relevant empirically supported laws. However, that this
is possible with hindsight is irrelevant to the question of what attitude
should be adopted towards inconsistent theories before such consistent
reconstructions have been identified (see Brown (1990), p. 292; Brown
(1993); Smith (1988a), and Smith (1988b)). An alternative strand within
this approach is to regard inconsistent theories as “proto-theories” and shift
away from a concern with truth to an epistemic attitude of “entertainment”
(Smith (1988a)). Inconsistency can then be accommodated by abandoning

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closure for such “proto-theories”, in the sense that the latter can be broken
up into (self-) consistent sub-sets from each of which implications can be
derived in classical fashion.
Abandoning closure in order to accommodate inconsistency is a well
known move, of course. Thus Kyburg has remarked that “it is not the strict
inconsistency of the rational corpus that leads to trouble – it is the imposition of deductive closure” (Kyburg (1987), p. 147). Smith, in particular,
follows Harman in claiming that accepting deductive closure should not
be confused with inference (Smith (1988a)). Harman in turn dismisses
the “Logical Closure Principle” on the grounds that closure is simply an
unrealistic requirement (see Harman (1986), pp. 12–14). Together with the
acknowledged defeasibility of the claim that inconsistency is to be avoided,
such arguments are adduced to support the more general line that “[. . .]
there is no clearly significant way in which logic is specially relevant to
reasoning” (ibid., p. 20).
What began as a maneuver to accommodate inconsistency while staying outside a paraconsistent framework has terminated in a position of such
extremity that one can only wonder if the supposed cure is worse than
the disease. Furthermore, without the imposition of closure, there seems
nothing to prevent the division into consistent sub-sets from becoming an
unsystematic affair, with “anything goes” being the order of the day with
regard to the application of claims from these sub-sets. Finally, talk of
“entertaining” Bohr’s model as “proto-theory” is inappropriate, given the
level of commitment maintained in its exploitation and further development. Such commitment is the hallmark of full blooded acceptance and
what is required is some way of doing justice both to this attitude and
the view that inconsistent theories should not be regarded as true in the
correspondence sense.
There is a great deal more to be said here, of course (see da Costa
and French (forthcoming)), but our claim is that “doing justice” to such
doxastic attitudes involves the elaboration of an alternative account of belief according to which “belief that p” is not to be understood as “belief
that p is true”, in the correspondence sense. Such an account has been
developed in da Costa and French (1989); (1993a); (1993b); and French
(1991), where the view is defended that, when it comes to representational
structures such as scientific theories, “belief that p” is to be understood
as “belief that p is pragmatically or partially true”. This allows for the
accommodation of inconsistency by acknowledging that it is not a permanent feature of reality to which theories must correspond, but is rather
a temporary aspect of such theories which may nevertheless be extremely
fruitful in a heuristic sense (see da Costa and French (1993a) and da Costa,

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Bueno and French (forthcoming)). On this account it is not the “logic of
science”, in the sense of the underlying logic of deduction and inference,
which is paraconsistent, but rather the appropriate “logic of truth”.
Relatedly, the logic of pragmatic truth as delineated above may serve as
a “logic of scientific acceptance” (see da Costa and French (1993a)). The
nature of acceptance is relatively little discussed within the philosophy of
science. Those accounts that do consider it tend to divide between two
extremes: those that identify acceptance and belief and those that separate
the two entirely. The former typically regard belief in terms of the correspondence view of truth, whereas the latter fall prey to the accusation of
conventionalism in theory choice (for further discussion, see da Costa and
French (1993a)). An alternative “via media” is to retain the connection
between belief and acceptance whilst rejecting truth-as-correspondence.
In this view, to accept a theory is to be committed, not to believing it to
be true per se, but to holding it as if it were true, for the purposes of further elaboration, development and investigation. Thus acceptance involves
belief that the theory is partially or pragmatically true only and this, we
believe, corresponds to the fallibilistic attitude of scientists themselves.7
Linking acceptance and pragmatic truth in this way restores a formal
similarity between “truth”, taken generally, and acceptance with regard to
deductive closure. It has been argued, for example, that acceptance differs
from truth in that whereas the latter is deductively closed, in the sense that
what one deduces from a set of truths is also true, the former generally
is not (see Ullian (1990)). This is correct if closure is understood only in
classical terms. However, what the above formal analysis shows is that
acceptance, understood within the framework of pragmatic truth, may be
regarded as closed under the Jaskowski’s discussive system (da Costa and
French (1993a)). To put it more precisely: although there is no closure
under classical conjunction and material implication, one can define discussive forms of implication and conjunction as above, with respect to
which acceptance can indeed be regarded as closed.
This is a result of both general and particular significance. Our contention is that inconsistency can be accommodated within an appropriate
framework under which the set of propositions which we accept is closed
under implication. The appropriate framework is precisely that which we
have presented in this paper and the form of implication is, of course, discussive. Shifting perspective again from the specific to the more general,
it is the failure to consider such non-classical systems which undercuts
the claim that “logic” is not specially relevant to reasoning. Within the
framework of pragmatic truth we can accommodate inconsistency while

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still retaining a sense of deductive closure. In this manner the relevance of
logic to reasoning – especially scientific reasoning – is restored.
1 The authors wish to acknowledge the helpful comments of two anonymous referees

for this journal.
2 Interesting enough, Colin Howson, at a recent meeting of the British Society for the
Philosophy of Science, in which one of us (S. French) was presenting a piece (French
(1997)), asked for the connections between the logic of pragmatic truth and Kripkean
semantics. Though this paper was already written when this meeting took place, it certainly
supplies an answer to Howson’s question.
3 More formally, an n-place partial relation R can be viewed as a triple hR , R , R i,
1 2 3
where R1 , R2 , and R3 are mutually disjoint sets, with R1 ∪ R2 ∪ R3 = D n , and such that
R1 is the set of n-tuples that belong to R; R2 the set of n-tuples that do not belong to R;
and finally R3 of those n-tuples for which it is not defined whether they belong or not to
R. (Note that when R3 is empty, R is a normal n-place relation that can be identified with
R1 .)
4 Related works in Jaskowski’s logic are: D’Ottaviano and da Costa (1970), D’Ottaviano
(1985), D’Ottaviano and Epstein (1988), Avron (1991), and Epstein (1995), Chapter IX.
5 It should be noticed that in our system the Barcan formula is a theorem. This comes,
in particular, from the fact that A-normal structures have the same universe (see Definition 1).
6 We assume, in this context, that there are essentially two negations. In paraconsistent
situations, we have to use the weak negation ¬; in classical situations, we may use the
strong negation ¬♦α which behaves in the same way as the classical negation of α. The
gist of our logic is to handle both negations according to our needs.
7 The formalism articulated here can also be used in defense of an empiricist interpretation of science, since it provides tools for an agnostic view about theoretical terms (as is
the case of van Fraassen’s constructive empiricism; see van Fraassen (1980) and (1989)).
Consider, for instance, the structures discussed in Section 1. Roughly speaking, we use
those structures with domain A1 to accommodate observable features of physical systems,
and those with domain A2 to describe non-observable, theoretical properties introduced
in the study of such systems. The set of accepted sentences P puts constraints on the
extensions of the pragmatic structures under consideration. In this context, a sentence α
is pragmatically true if it is true (in the Tarskian sense) under one possible assignment of
values to the theoretical terms which agrees with the accepted information in P and with
the observable phenomena in A1 . This assignment is done, of course, in a convenient Anormal structure. Further discussion about this issue can be found in da Costa and French
(1990), Bueno (1996) and Bueno (1997).

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Department of Philosophy,
University of São Paulo,
São Paulo-SP, 05508-900, Brazil
(E-mail: ncacosta@usp.br)
Division of History and Philosophy of Science,
Department of Philosophy,
University of Leeds,
Leeds, LS2 9JT, UK
(E-mail: phloab@leeds.ac.uk)

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